Yes, you can do this with one flow, but you cannot do it with the For A Selected Item trigger. Instead use the HTTP Request trigger.
This trigger provides a public URL, with which you can trigger the flow. You can use column formatting in sharepoint to pass the site, document library and ID of the item. With this information you can then get the relevant document in SharePoint. So instead of creating 15+ flow, you only need to change the column formatting on each site.
There are 2 caveats here:
1. The URL is public. So anyone with on the internet can potentially trigger your flow, if they have the URI.
2. The user will be forwarded to a new page, because they clicked on a link. You could have the flow display a message like: "Copying in progress."
Best Regards
---------------------------------------------------
If my answer solved your problem, please mark it as a solution.
If what I wrote helped you, please give it a thumbs up.